The popular MO question "Famous mathematical quotes" has turned
up many examples of witty, insightful, and humorous writing by
mathematicians. Yet, with a few exceptions such as Weyl's "angel of
topology," the language used in these quotes gets the message
across without fancy metaphors or what-have-you. That's probably the
style of most mathematicians.

Occasionally, however, one is surprised by unexpectedly colorful
language in a mathematics paper. If I remember correctly, a paper of
Gerald Sacks once described a distinction as being

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59

Latest paper, my co-author put in "but we will choose a more painful way, because there is nothing like pain for feeling alive" but the referee jumped on it.
–
Will JagyApr 23 '10 at 5:09

16

Maybe I should expand the question to include colorful language cut from serious mathematics papers :)
–
John StillwellApr 23 '10 at 5:18

33

By the way, your remark reminds me of another in a similar spirit that made it into the Princeton Companion. In his article on algebraic geometry, János Kollár says of stacks: "Their study is strongly recommended to people who would have been flagellants in earlier times."
–
John StillwellApr 23 '10 at 7:49

30

I was actually rather surprised recently by a referee who did not know the phrase “red herring”, and had to look it up. He insisted that we change it to something more understandable. It makes me wonder how much “colourful” language is weeded out by referees, and whether the mathematical literature is poorer for it.
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Harald Hanche-OlsenApr 24 '10 at 2:31

26

@Harald: If you intend your mathematical papers to be read by a wide range of readers, then write them in simple language, suitable for those who are relative beginners in English. I remember reading long ago some metaphoric phrase in a mathematics research paper, then imagining students all over the world getting out their English dictionaries, looking it up, and still not understanding what it meant. (I no longer remember what the phrase was, just this reaction to it.)
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Gerald EdgarApr 24 '10 at 15:43

You'll find a whole host of colourful language and allusions scattered throughout the works of Kato. To quote just one example from his Lecture on the approach to Iwasawa theory for Hasse Weil L-functions via $B_{dR}$:

Where is the homeland of zeta values to which the true reasons of celestial phenomena of zeta values are attributed ? How can we find a galaxy train to approach it, which runs through the galaxy of p-adic zeta elements and whose engine is the theory of p-adic periods ? I imagine that one coach of the train has the name 'explicit reciprocity law of p-adic Galois representations'.

Kato lectures like this too. His lectures (at least in the 1990s) often used to start with various bits of philosophy of this nature. I remember vividly his explaining at the IAS that the reason Bloch and Beilinson constructed the right zeta elements in K_2 was that they had very large mouths and loved their wives (and then a long explanation of why these things were relevant, which unfortunately this margin won't contain). It wouldn't surprise me if these comments ended up in print at some point---that's Kato.
–
Kevin BuzzardApr 24 '10 at 8:16

1

Kato teaches like this as well. I remember him teaching theta functions, circa 2004, and coming through as a member of some strange cult (to me at least). Lots of mysticism, lots of references to the occult and kabbala and how the theta function is part of some spiritual realm, and the search for the "true theta function".
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Daniel MoskovichApr 29 '10 at 4:43

The paper "Division by three" by Peter Doyle and John Conway has a wealth of colorful language including:

"If the arrows are good, straight, American arrows, it is very natural for each arrow to dream of marrying the arrow next door."

and

"Not that we believe there really are any such things as inﬁnite sets, or that
the Zermelo-Fraenkel axioms for set theory are necessarily even consistent.
Indeed, we’re somewhat doubtful whether large natural numbers (like $80^{5000}$
, or even $2^{200}$) exist in any very real sense, and we’re secretly hoping that
Nelson will succeed in his program for proving that the usual axioms of
arithmetic—and hence also of set theory—are inconsistent. (See Nelson [6].)
All the more reason, then, for us to stick with methods which, because of their
concrete, combinatorial nature, are likely to survive the possible collapse of
set theory as we know it today."

Yes, although is good to remember that this is an unpublished manuscript, and that Conway "has never approved of this exposition, which he regards as full of fluff." I think this paper would benefit itself immensely if 20 or so pages were left out.
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Andres CaicedoMay 16 '10 at 14:39

The reader who makes it to the later chapters of M. N. Huxley's Area, Lattice Points and Exponential sums is rewarded with the following gem:

"If mathematics were an orchestra, the exponentials would be the violins. The $\rho(t)$ would be the flutes; they are introduced by the exponentials. The Poisson summation formula would be the tuba: powerful, but ridiculous when used too much"

P. T. Johnstone's On a Topological Topos has some interesting choices of words. Sometimes the words are discussed in parenthetical notes.

([...] we are tempted also to introduce the term 'consequential space' for an arbitrary object of $\mathcal{E}$, apart from a slight reluctance to give the name 'space' to an object of a category whose underlying-set functor is not faithful—and, we must admit, the fear that somebody will at once invent a notion of 'inconsequential space'.)

Sometimes there is no more than a reference to existing literature.

The rest of the proof of Theorem 5.1 is a fairly straightforward woozle-hunt (Milne [27])

Reference [27] is, as you may have guessed, A. A. Milne's Winnie The Pooh.

isn't there some humorous intention in the choice of the name "pointless topology" ?
–
Pietro MajerJan 17 '11 at 23:54

32

+1 for Pooh, but: It should be remarked that Woozle-Hunting is a rather poor proof technique, given that it involves going in circles for a Long Time, and ends without capturing any Woozles at all. In fact, a proof by Woozle-hunt (that actually proved something) would be a remarkable achievement.
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Ketil TveitenFeb 4 '11 at 9:27

Frank Adams was notorious for slipping little gems of humour into his paper and books.
For instance, from his book, "Infinite Loop Spaces,"

(p. 128)

The reader may expect me to say something about "double coset fomulae." I shall indeed; I advise you to avoid them.

(p. 131)

Of course, this still leaves the question: what do you say to the algebraist who loves double cosets and insists that this is the same thing really? I suggest that you smile politely and say that you are maximizing your chance of finding a helpful and congenial interpretation of the double cosets. There is no need to say that the best interpretation is one which allows you to avoid mentioning the (expletive deleted) things at all.

For further entertainment, look at the entry [85] in the bibliograph, and look at "jokes" in the index.

Yes, I disagree with Adams about double cosets. Then again the colourful stories about him that came out after he died seemed to me to indicate that I disagreed with him about a number of things (for example the merits of attacking people with axes)
–
Kevin BuzzardApr 24 '10 at 8:13

13

@Kevin: Hmm, I don't know which side of the axe issue you stand on. You know what -- it'll come up eventually. Why don't you surprise me?
–
Pete L. ClarkApr 24 '10 at 15:23

5

I hope this is not seen as mean-spirited, but some years ago, once when I mentioned Adams over coffee (probably in the context of his Lie Groups book) someone asked if I'd heard the joke about the "unstable Adams spectral sequence". (It tickled my fancy; everyone else went back to talking about traffic or football.)
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Yemon ChoiApr 25 '10 at 0:54

A more imaginative nomenclature than one relying on overburdened terms such as "fundamental," "principal," "regular," "normal," "characteristic," "elementary," and so on is desirable. Inventors of important mathematical notions should give their inventions suggestive names. The disadvantage that good names might prevent the inventor's name from being immortalized as an adjective would be more than compensated by the advantage that this honor could not possibly be bestowed on noninventors.

I was always amazed that Clifford Truesdell could get away with a quote like this:

Nowadays, when the common student
seeks a secure berth by grafting
himself upon some modest little
professor whom he regards as prone to
foster painlessly his limaceous glide
toward a dissertation not too
strenuous or, even better, to
draught it for him, tradition is
moribund (...)

This is from his introduction to the selected papers of W. Noll. Admittedly, Truesdell was the chief editor himself, and could write therefore whatever he wanted, but it's still pretty strong. Felt too close to home when I first read it as a graduate student!

Truesdell is also the author of the single best Math Review ever: "In this paper are presented incorrect solutions to trivial problems. The basic error, however, is not new."
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Allen KnutsonApr 25 '10 at 16:05

To be nitpicky, the quotation is not quite right. The exact words are "This paper, whose intent is stated in its title, gives wrong solutions to trivial problems. The basic error, however, is not new: [...]."
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Hans LundmarkJul 1 '10 at 8:06

14

I believe that Bass gets credit for a book review with the line- "this book fills a much needed gap in the literature".
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aginenskyJan 12 '11 at 2:52

5

@aginesky: Actually the "much needed gap" is due to my colleague Lee Neuwirth. He put it in a review that the wrote either as a grad student or recent post-doc. Ralph Fox (his advisor) read it and roared with laughter. It was excised from the published version, but quickly made the rounds.
–
Victor MillerJan 16 '11 at 16:03

I must post some more examples from Frank Adams. I recommend reading the last section of his paper "Finite H-Spaces and Lie Groups", which contains a letter to the reader written in the voice of the exceptional lie group E8. Two excerpts :

"This is as if one were to award a title for drinking beer, having first fixed the rules so as to exclude all citizens of Heidelberg, Munich, Burton-on-Trent, and any other place where they actually brew or drink much of the stuff."

"In the second place, to consider the question at all reveals a certain preoccupation with ordinary cohomology. Any impartial observer must marvel at your obsession with this obscure and unhelpful invariant."

"Unfortunately, it appears that there is now in your world a race of
vampires, called referees, who clamp down mercilessly upon mathematicians
unless they know the right passwords. I shall do my best to
modernize my language and notations, but I am well aware of my shortcomings
in that respect ; I can assure you, at any rate, that my intentions
are honourable and my results invariant, probably canonical, perhaps
even functorial.But please allow me to assume that the characteristic is
not 2"

Jon Barwise's Admissible Sets and Structures contains the following on page 69:

When used in a class or seminar, section 6 should be supplemented with coffee (not decaffeinated) and a light refreshment. We suggest Heatherton Rock 'Cakes. (Recipe: Combine 2 cups of self-rising flour with 1 t. allspice and a pinch of salt. Use a pastry blender or two cold knives to cut in 6 T butter. Add 1/3 cup each of sugar and raisins (or other urelements). Combine this with 1 egg and
enough milk to make a stiff batter (3 or 4 T milk). Divide this into 12 heaps, sprinkle with sugar, and bake at 400 °F. for 10 — 15 minutes. They taste better than they sound.)

There is a response to this (with stronger ingredients) somewhere in Aki Kanamori's The Higher Infinite but I forgot exactly where. Later in that book, on page 289, Kanamori writes:

But first, a respite from the rigors: Instead of yet another recipe, we offer the following chess problem (M. Henneberger, first and second prize, "Revista de Sah" 1928):

"Indeed, I wrote the outline of this book while wandering across India, so that, in my mind, Henkin's method is inexorably linked to the droves of wild elephants I met while crawling among the swamp plants of the preserves of Kerala; the elimination of imaginaries, to the gliding vultures above the high Himalayan peaks; and the theorem of the bound, to the naked bodies of the Mauryan women that the traveler saw on the bends of a jungle trail, before they had time to cover themselves. I dare hope only that this book will evoke similarly pleasant images in my reader; I wish it will be as pleasant a companion for you as it was for me."

From Bruno Poizat's "Model Theory". He also constantly belittles the readers of the English edition of the book. Highly recommended!

He was inspired by the following famous UK comic: (en.wikipedia.org/wiki/The_Fat_Slags) I saw him give a talk on the subject once. When the phrase came up all the English people in the audience laughed and everyone else looked around with very confused expressions on their faces.
–
Joel FineApr 24 '10 at 8:14

3

This is more colloquial than you think! The Fat Slags are a pair of well-known cartoon characters from Viz magazine. Given that he's a Brit, it's surely a reference to them.
–
Kevin BuzzardApr 24 '10 at 8:20

I hate to comment like this, but for community wiki, I feel less bad, and it won't affect the ranking at this moment...could someone vote this one up? I'd rather like it to have 27 votes, but no more. It feels fitting.
–
Charles SiegelJun 2 '10 at 13:07

4

Now we just have to get your comment to 27 upvotes as well.
–
Steven GubkinJun 8 '10 at 13:05

Two from Casselman's "A companion to Macdonald's book on p-adic spherical functions":

The word ‘´epingler’ means ‘to pin’,
and the image that comes to mind most
appropriately is that of a mounted
butterfly specimen. [Kottwitz:1984]
uses ‘splitting’ for what most call
‘´epinglage’, but this is not
compatible with the common use of
‘deploiement’, the usual French term
for ‘splitting’.) Ian Macdonald, among
others, has suggested that retaining
the French word ´epinglage in these
notes is a mistake, and that it should
be replaced by the usual translation
‘pinning.’ This criticism is quite
reasonable, but I rejected it as
leading to noncolloquial English. The
words ‘pinning’ as noun and ‘pinned’
as adjective are commonly used only to
refer to an item of clothing worn by
infants, and it just didn’t sound
right.

and

These phenomena are part of what
Langlands calls endoscopy, a word that
might be roughly justified by saying
that endoscopy is concerned with some
fine aspects of the structure of
harmonic analysis on a reductive
p-adic group. Langlands attributes the
term to Avner Ash, praising his
classical knowledge, but I was pleased
to find recently the following
quotation that shows a more vulgar
intrusion of endoscopy into the modern
world:

Jeeves: “ . . . I had no need of the
endoscope.”

Bertie: “The what?”

Jeeves: “Endoscope, sir. An instrument
which enables one to peer into the . .
. interior and discern the core.”

From Chapter 12 of Jeeves and the
feudal spirit by P. G. Wodehouse.

This discussion is about distingishing
fae jewlry from real. Since the
endoscope also has medical uses, one
could imagine an even more vulgar
usage.

He has modified the notes several times so these might not be there anymore, but I have the older copies =)

My girlfriend is a surgeon and once a month our copy of "Endoscopy" drops through the post box. I tried to out-do her recently by sitting on the sofa reading a paper of Waldspurger about "twisted endoscopy" and she suggested he was doing it wrong.
–
Kevin BuzzardApr 24 '10 at 8:22

4

You made the effort, that's what counts in the end.
–
Will JagyApr 24 '10 at 19:05

I don't even know if this is intentional or not. In his book Teichmuller theory, John Hubbard frequently references the category of Banach Analytic Manifolds. He adheres to the convention that a category be referenced by the concatenation of the first three letters of each constituent word, making the category in question BanAnaMan. This still cracks me up to this day.

Heh. I am sure this was discovered by coincidence and kept by design.
–
Yemon ChoiApr 25 '10 at 0:49

4

Greg, congratulations on the great answer badge!
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John StillwellSep 17 '10 at 18:06

7

Greg, I've just had a look at Hubbard's Teichmüller Theory, and a wonderful book it is. But, alas, I think your memory has deceived you, because his abbreviation for the category of Banach analytic manifolds (page 165) is in fact BanMan.
–
John StillwellMar 28 '11 at 23:31

10

Hmm, since my observation came from a course with Prof. Hubbard using a preprint of the book, I guess he changed it before publication. Thats a little disappointing.
–
Greg MullerMar 29 '11 at 16:27

5

It has belatedly struck me that there should really be a contravariant equivalence of categories between Ban(Ana)Man and some category of algebraic objects, which could be abbreviated to ERIC.
–
Yemon ChoiAug 23 '11 at 0:56

At the risk of blowing my own horn, I will mention the line in the book, Category Theory for Computing Science" by Charles Wells and me. After mentioning the Russell paradox and how to avoid it, we say, "This prophylaxis guarantees safe sets." I caught at least one colleague rolling on the floor laughing, but only after reading it aloud.

This is perhaps more of a silly play on words than colourful, but I still got a laugh out of it. One page 58 of Conway's 'The sensual (quadratic) form' while discussing Kneser's gluing method a sentence begins:

I have the book but I don't get the pun, and I feel the lesser for it. Could you please explain it, if not in comments or answers here then, say, in your MO "profile" autobiography field or in email to me?
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Will JagyApr 24 '10 at 4:08

2

It's likely that I have a very dry sense of humour. But, if Conway was being formal he would write "To further illuminate the utility of the gluing method,..". I can't help but feel that it is written the way it is quite deliberately.
–
Robby McKilliamApr 24 '10 at 21:48

2

I think I see, and I agree that it was deliberate. I was looking for song titles that rhymed, as "Cupidity Fondue," "Venality of You," "Morality Imbue."
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Will JagyApr 24 '10 at 22:50

In this MO answer, I mentioned Arnold Miller's lecture
notes, where he gives
an entertaining account of the MM proof system (for Micky Mouse), having as axioms all validities and modus ponens as the only rule of inference. Although it is easy to prove the Completness theorem from Compactness in this system, it is nevertheless a kind of joke system, since the set of validities is not a decidable set, and so we would be fundamentally unable to recognize whether something is a proof or not in this system. Miller uses this example to illustrate the point as follows:

The poor MM system went to the Wizard of OZ and said, “I
want to be more like all the other proof systems.” And the
Wizard replied, “You’ve got just about everything any other
proof system has and more. The completeness theorem is easy
to prove in your system. You have very few logical rules
and logical axioms. You lack only one thing. It is too hard
for mere mortals to gaze at a proof in your system and tell
whether it really is a proof. The difficulty comes from
taking all logical validities as your logical axioms.” The
Wizard went on to give MM a subset Val of logical
validities that is recursive and has the property that
every logical validity can be proved using only Modus
Ponens from Val.

And he then goes on to describe how one might construct
Val, and give what amounts to a traditional proof of
Completeness.

There is the famous (and with contradictory interpretations) cry from Jean Dieudonné "à bas Euclide !", "Down with Euclide !". His books and prefaces are good sources for strong (and dated) opinions on what was "good" or "productive" mathematics and what was not.

S. S. Abhyankar's book, "Algebraic Geometry for Scientists and Engineers" is actually more for mathematicians, and algebraic geometers in particular. It has the following quip(meant for Andre Weil who wanted to eliminate elimination theory):

From Strichartz's A Guide to Distribution Theory and Fourier Transforms:

(p.2) "You have almost seen the entire definition of generalized functions. All you are lacking is a description of what constitutes a test function and one technical hypothesis of continuity. Do not worry about continuity--it will always be satisfied by anything you can construct (wise-guys who like using the axiom of choice will have to worry about it, along with wolves under the bed, etc)."

This is a little off the mark (from a textbook), but Exercise VIII.8.3 of Sarason's [Notes on] Complex Function Theory is:

Stand straight with feet about one
meter apart, hands on hips. Bend at
the waist, knees straight, and touch
left foot with right hand.
Straighten. Bend again and touch
right foot with left hand.
Straighten. Repeat 15 times.

I was a TA on a course taught by Sarason himself following this book. I had two students "solve" that Exercise during one of my office hours.
–
Alfonso Gracia-SazMay 6 '10 at 5:02

14

This reminds me of an exercise (C.67) from the chapter about linear transformations in Peter Hackman's Linear Algebra textbook Kossan (mai.liu.se/~pehac/titta/kossaboken). It's in Swedish, but here's an attempt to translate it: "Seize the ends of a pointer between your extended arms, and turn yourself an angle of $v$ radians about your own vertical axis. What have you proved then? Try the same maneuver with the pointer in your right hand, aligned with your straigh arm. Show this to someone who has never studied Linear Algebra. Interpret the result."
–
Hans LundmarkJul 1 '10 at 8:25

6

This reminds me of one of Professor Imre Leader's example sheets, which features the question "what can you infer from the previous question about the lecturer's ability to typeset matrices?". Another one, interspersed with serious questions asking for proofs of various equivalences involving the well-ordering principle, was "what's yellow and equivalent to the axiom of choice?".
–
Adam P. GoucherJun 28 '14 at 17:40

Upon a preliminary perusal, this parable may appear to be about pairs of palindromes, periods, and pitiful alliteration. In actuality, however, it is the story of a real quadratic irrational number $\alpha$ and its long-lost younger sibling, its algebraic conjugate $\tilde{\alpha}$ ($\alpha > \tilde{\alpha}$). How in the dickens are all these notions connected? We begin at the beginning...
Although the conjugates $\alpha$ and $\tilde{\alpha}$ are not *identical* twins, unlike the two zeros of $(x - 3)^2$, they do share a common family history: they each were born of the same irreducible parent polynomial having rational coefficients,
$$P_{\alpha}(x) = P_{\tilde{\alpha}}(x) = (x - \alpha)(x - \tilde{\alpha}) = x^2 - \text{Trace}(\alpha)x + \text{Norm}(\alpha),$$
where $\text{Trace}(\alpha) = \alpha + \tilde{\alpha}$ and $\text{Norm}(\alpha) = \alpha \tilde{\alpha}$. Perhaps not surprisingly, some conjugate pairs exhibit similar personalities. But how similar can they be? And how can we detect those similarities simply by looking at $\alpha$? As we will discover as our tale unfolds, the answer - foreshadowed in the title - is encoded in what can be described as the number theoretic analogue of the DNA-sequence for $\alpha$. However, before delving into $\alpha$'s genes, we first motivate our results by weaving a lattice of algebra.

The English translation by Kenji Iohara of Minoru Wakimoto's "Infinite dimensional Lie algebras" is as colourful as it gets, I think. For example on page 8

Namely, we can think of an element of U(A) as an element of A. But since U(A)and A are not isomorphic, this thinking is not an identification but a lonely unrequited love.

Or on page 26

An elegant shape of the left half of Mt. Fuji reflected in the surface of a lake, this is the proportion of the finite-dimensional representations of $\mathfrak{sl}(2,\mathbb{C})$.

Or on page 27

Since ancient times, it has been the charm of music that has soothed the fiercest warriors (or samurai). This law seems to be universal in the physical universe, and it is also true in the world of Lie algebras.

My personal favourite is on page 289

Moreover, the conformal superalgebra (CSA for short) has recently been discovered by Kac, and its definition is given in 2.7 of [K5]. This representation theory has been started in [CK], It is like a matsutake mushroom derived from a big tree called a vertex operator algebra, and it is a portable version of a super-conformal algebra and a vertex operator algebra. There is an experimental report saying that it is more delicious to munch a matsutake mushroom than its landlord- i.e. a Japanese red pine.Let us munch it a bit.

Unfortunately perhaps, the language is not nearly as colourful in the original Japanese (it's just an outstandingly good book), and is an artifact of the translation. I've long had a dream of doing a more sober translation... but I suppose that Iohara's translation is not without its charm. Anyway, the colourful language is in my opinion is to be attributed to Iohara rather than to Wakimoto.

@Mariano I've read the original, which is an outstanding book BTW, but the colourfulness is pretty much (90% at least) Iohara's in my opinion. Or perhaps, it sounds better in Japanese (the translator isn't making stuff up, but he certainly makes it sound more colourful than it was). Contrary to what my appearance might suggest, I speak and read Japanese fluently.
–
Daniel MoskovichApr 29 '10 at 7:19

I can't say whether this is more than Freudian typo, but I know of another Freudian typo that nearly got into print. When Springer was preparing the 2nd edition of my book on topology and combinatorial group theory they sent me (in all seriousness) a copy of the intended new cover with the title Classical Topology and Combinatorial Group Therapy.
–
John StillwellApr 30 '10 at 21:26

5

Seeing that it is in quotes, I bet it is an intentional pun on s_ymplectic r_eduction.
–
Willie WongMay 3 '10 at 9:37